Methods | Part IB, 2004

The equation governing small amplitude waves on a string can be written as

2yt2=2yx2\frac{\partial^{2} y}{\partial t^{2}}=\frac{\partial^{2} y}{\partial x^{2}}

The end points x=0x=0 and x=1x=1 are fixed at y=0y=0. At t=0t=0, the string is held stationary in the waveform,

y(x,0)=x(1x) in 0x1.y(x, 0)=x(1-x) \quad \text { in } \quad 0 \leq x \leq 1 .

The string is then released. Find y(x,t)y(x, t) in the subsequent motion.

Given that the energy

01[(yt)2+(yx)2]dx\int_{0}^{1}\left[\left(\frac{\partial y}{\partial t}\right)^{2}+\left(\frac{\partial y}{\partial x}\right)^{2}\right] d x

is constant in time, show that

n odd n11n4=π496\sum_{\substack{n \text { odd } \\ n \geqslant 1}} \frac{1}{n^{4}}=\frac{\pi^{4}}{96}

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